76.12.8 problem Ex. 8

Internal problem ID [20058]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 8
Date solved : Thursday, October 02, 2025 at 05:17:24 PM
CAS classification : [[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 2 x -y+1+\left (2 y-x -1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.127 (sec). Leaf size: 33

ode:=2*x-y(x)+1+(2*y(x)-x-1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\sqrt {4-27 \left (x +\frac {1}{3}\right )^{2} c_1^{2}}+\left (3 x +3\right ) c_1}{6 c_1} \]

✓ Mathematica. Time used: 0.082 (sec). Leaf size: 67

ode=(2*x-y[x]+1)+(2*y[x]-x-1)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (-i \sqrt {3 x^2+2 x-1-4 c_1}+x+1\right )\\ y(x)&\to \frac {1}{2} \left (i \sqrt {3 x^2+2 x-1-4 c_1}+x+1\right ) \end{align*}

✓ Sympy. Time used: 1.641 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x + (-x + 2*y(x) - 1)*Derivative(y(x), x) - y(x) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {x}{2} - \frac {\sqrt {C_{1} - 27 x^{2} - 18 x}}{6} + \frac {1}{2}, \ y{\left (x \right )} = \frac {x}{2} + \frac {\sqrt {C_{1} - 27 x^{2} - 18 x}}{6} + \frac {1}{2}\right ] \]